# Volume 54, № 5, 2002

### Oleksandr Ivanovych Stepanets' (on his 60-th birthday)

Lukovsky I. O., Makarov V. L., Mitropolskiy Yu. A., Romanyuk A. S., Romanyuk V. S., Rukasov V. I., Samoilenko A. M., Serdyuk A. S., Shevchuk I. A., Zaderei P. V.

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 579-580

### Approximation of Convolution Classes by Fourier Sums. New Results

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 581-602

We present a survey of new results related to the investigation of the rate of convergence of Fourier sums on the classes of functions defined by convolutions whose kernels have monotone Fourier coefficients.

### Kolmogorov-Type Inequalities for Periodic Functions Whose First Derivatives Have Bounded Variation

Babenko V. F., Kofanov V. A., Pichugov S. A.

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 603-609

We obtain a new unimprovable Kolmogorov-type inequality for differentiable 2π-periodic functions *x* with bounded variation of the derivative *x*′, namely $$\left\| {x'} \right\|_q \leqslant K\left( {q,p} \right)\left\| x \right\|_p^a \left( {\mathop V\limits_{0}^{{2\pi }} \left( {x'} \right)} \right)^{1 - {alpha }} ,$$ where *q* ∈ (0, ∞), *p* ∈ [1, ∞], and α = min{1/2, *p*/*q*(*p* + 1)}.

### Criterion of Polynomial Denseness and General Form of a Linear Continuous Functional on the Space $C_w^0$

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 610-622

For an arbitrary function $w:\mathbb{R} \to \left[ {0,1} \right]$, we determine the general form of a linear continuous functional on the space $C_w^0$. The criterion for denseness of polynomials in the space $L_2 \left( {\mathbb{R},d\mu } \right)$ established by Hamburger in 1921 is extended to the spaces $C_w^0$.

### Generalized Moment Representations and Padé Approximants Associated with Bilinear Transformations

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 623-627

By using the method of generalized moment representations with an operator of bilinear transformation of an independent variable, we construct elements of the first subdiagonal of the Padé table for certain special power series.

### On Modified Strong Dyadic Integral and Derivative

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 628-638

For functions *f* ∈ *L*(*R* _{+}), we define a modified strong dyadic integral *J*(*f*) ∈ *L*(*R* _{+}) and a modified strong dyadic derivative *D*(*f*) ∈ *L*(*R* _{+}). We establish a necessary and sufficient condition for the existence of the modified strong dyadic integral *J*(*f*). Under the condition \(\smallint _{R_ + }\) *f*(*x*)*dx* = 0, we prove the equalities *J*(*D*(*f*)) = *f* and *D*(*J*(*f*)) = *f*. We find a countable set of eigenfunctions of the operators *J* and *D*. We prove that the linear span *L* of this set is dense in the dyadic Hardy space *H*(*R* _{+}). For the functions *f* ∈ *H*(*R* _{+}), we define a modified uniform dyadic integral *J*(*f*) ∈ *L* ^{∞}(*R* _{+}).

### On the Convergence of Fourier Series in the Space $L_1$

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 639-646

We establish necessary and sufficient conditions for the convergence in the mean of trigonometric series whose coefficients satisfy the Boas–Telyakovskii conditions.

### Approximation of Sobolev Classes by Their Sections of Finite Dimension

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 647-655

For Sobolev classes of periodic functions of one variable with restrictions on higher derivatives in *L* _{2}, we determine the exact orders of relative widths characterizing the best approximation of a fixed set by its sections of given dimension in the spaces *L* _{q}.

### (ϕ, α)-Strong Summability of Fourier–Laplace Series for Functions Continuous on a Sphere

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 656-665

We establish upper bounds for approximations by generalized Totik strong means applied to deviations of Cezàro means of critical order for Fourier–Laplace series of continuous functions. The estimates obtained are represented in terms of uniform best approximations of continuous functions on a unit sphere.

### On One-Sided Approximation of Functions with Regard for the Location of a Point on an Interval

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 666-669

We investigate a pointwise approximation of functions of the class *H* ^{ω} (ω(*t*) is a modulus of continuity convex upward) by absolutely continuous functions with variable smoothness.

### Approximation of Classes $B_{p,θ}^r$ by Linear Methods and Best Approximations

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 670-680

We investigate problems related to the approximation by linear methods and the best approximations of the classes $B_{p,{\theta }}^r ,\; 1 ≤ p ≤ ∞$ in the space $L_{∞}$.

### Approximation of the Classes $C^{{\bar \psi }} H_{\omega }$ by de la Vallée-Poussin Sums

Chaichenko S. O., Rukasov V. I.

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 681-691

We investigate the problem of the approximation of the classes $C^{{\bar \psi }} H_{\omega }$ introduced by Stepanets in 1996 by the de la Valée-Poussin sums. We obtain asymptotic equalities that give a solution of the Kolmogorov–Nikol'skii problem for the de la Valée-Poussin sums on the classes Cψ¯HωCψ¯Hω in several important cases.

### Approximation of Periodic Analytic Functions by Interpolation Trigonometric Polynomials in the Metric of the Space $L$

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 692-699

We obtain asymptotic equalities for the upper bounds of approximations by interpolation trigonometric polynomials in the metric of the space *L* on classes of convolutions of periodic functions admitting a regular extension into a fixed strip of the complex plane.

### Trigonometric Widths of the Classes $B_{p,θ}^{Ω}$ of Periodic Functions of Many Variables

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 700-705

We obtain exact order estimates for the trigonometric widths of the classes BΩp,θBp,θΩ of periodic functions of many variables in the space $L_{q, 1}\; < p ≤ 2 ≤ q < p/(p − 1)$.

### Approximation of Cauchy-Type Integrals

Savchuk V. V., Stepanets O. I.

Ukr. Mat. Zh. - 2002. - 54, № 5. - pp. 706-740

We investigate approximations of analytic functions determined by Cauchy-type integrals in Jordan domains of the complex plane. We develop, modify, and complete (in a certain sense) our earlier results. Special attention is given to the investigation of approximation of functions analytic in a disk by Taylor sums. In particular, we obtain asymptotic equalities for upper bounds of the deviations of Taylor sums on the classes of ψ-integrals of functions analytic in the unit disk and continuous in its closure. These equalities are a generalization of the known Stechkin's results on the approximation of functions analytic in the unit disk and having bounded *r*th derivatives (here, *r* is a natural number).

On the basis of the results obtained for a disk, we establish pointwise estimates for the deviations of partial Faber sums on the classes of ψ-integrals of functions analytic in domains with rectifiable Jordan boundaries. We show that, for a closed domain, these estimates are exact in order and exact in the sense of constants with leading terms if and only if this domain is a Faber domain.